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1 Bayesian Dynamic Factor Models and Variance Matrix Discounting for Portfolio Allocation Omar Aguilar and Mike West ISDS, Duke University, Durham, NC January 1998 Abstract We discuss the development of dynamic factor models for multivariate ænancial time series, and the incorporation of stochastic volatility components for latent factor processes. Bayesian inference and computation is developed and explored in a study of the dynamic factor structure of daily spot exchange rates for a selection of international currencies. The models are direct generalisations of univariate stochastic volatility models, and represent speciæc varieties of models recently discussed in the growing multivariate stochastic volatility literature. We also discuss connections and comparisons with the much simpler method of dynamic variance discounting that, for over a decade, has been a standard approach in applied ænancial econometrics in the Bayesian forecasting world. We review empirical ændings in applying these models to the exchange rate series, including aspects of model performance in dynamic portfolio allocation. We conclude with comments on the potential practical utility of structured factor models and future potential developments and model extensions. Keywords: Dynamic Factor Analysis; Dynamic Linear Models; Exchange Rates Forecasting; Markov Chain Monte Carlo; Multivariate Stochastic Volatility; Portfolio Selection; Variance Matrix Discounting. The authors acknowledge useful discussions with Jose M Quintana, Neil Shephard and Hong Chang, and partial support from NSF grants DMS and DMS This manuscript represents a preliminary draft report subject to revision. Before quoting or referencing, please check the authors' web site for a possible updated version. The latest version will be found as ISDS Discussion Paper on the Duke web site, 0

2 1 Introduction Since the mid-1980s the method of varianceècovariance discounting èquintana and West 1987, 88è has been used as a component of applied Bayesian forecasting models in ænancial econometric settings èquintana 1992; Putnam and Quintana 1994, 1995; Quintana and Putnam 1996; Quintana, Chopra and Putnam 1995è. In more recent years major developments in structured stochastic volatility èsvè modelling have led to the introduction of various approaches to modelling dependencies in volatility processes that, in principle, may lead to improvements in short-term forecasting of multiple ænancial and econometric time series. The potential is there for real improvements in practical short-term forecasting relative to the purely ëreactive" variance discounting methods that simply allow for volatility changes but do not anticipate the forms of change. Our interest here is to explore dynamic factor models as a context for multivariate stochastic volatility modelling, motivated by: æ questions about the potential for dynamic factor models to provide practical improvements in short-term forecasting, and resulting dynamic portfolio allocations, of international exchange rates and other ænancial time series; æ issues of model structuring, implementation, Bayesian analysis and computation; and æ questions of comparison with the simpler methods of ëtracking" volatility changes based on varianceècovariance discounting. We study these issues in connection with data analysis and portfolio construction using multiple series of international exchange rates. Variants of the basic method of variance matrix discounting èsee above references to Quintana and coauthorsè have formal theoretical bases in matrix-variate ërandom walks" èuhlig 1994, 97è. Further discussion is given below, and more background can be found in West and Harrison è1997, section è. The basic discounting methods follow foundational developments for univariate series in Ameen and Harrison è1995è and Harrison and West è1987è, and the formal multivariate models are direct generalisations of univariate models of Shephard è1994aè. Related discussion appears in West and Harrison è1997, section è. In the general multivariate context, the approach leads to the embedding of exponentially smoothed estimates of ëlocal" varianceècovariance structure within a Bayesian modelling framework, and so provides for adaptation to stochastic changes as time series data are processed. Modiæcations to allow forchanges in discount rates in order to adapt to varying degrees of change, including markedèabrupt changes in volatility patterns, extend the basic approach. The resulting update equations for sequences of estimated volatility matriceshave univariate components that relate closely to variants of ARCH and SV models, and so it is not surprising that they have proven useful in many applications. However, unlike these more formal models, discounting methods do not have real predictive capabilities, simply allowing for and estimating changes rather than anticipating them. Hence the interest in factor models that set out to explicitly describe changes through patterns of time-variation in parameters driving underlying latent processes. This is the key motivating concept underlying interest in SV models generally, and led has to various authors mentioningordeveloping multivariate SV models with dynamic factor structure. Key references include the initiating work of Harvey, Ruiz and Shephard è1994è, and the later 1

3 developments in the papers of Jacquier, Polson and Rossi è1994, 95è, and Kim, Shephard and Chib è1998è. In the following section we detail the basic framework and notation for stochastically time-varying variance matrices, and this leads into the basic factor structure as discussed by the above authors. This is followed by discussion of factor structure where we build on prior work in Bayesian factor analysis per se, adopting the structuring used by Geweke and Zhou è1996è in particular, and various extensions of basic factor models. We then discuss model ætting and computation, followed by analyses of international exchange rate time series with comparisons of the dynamic factor models with varianceècovariance discounting. The paper concludes with some summary and concluding comments. 2 Time Varying Variance Matrices and Factor Structure 2.1 Bayesian Discount Estimation To introduce notation, consider a q,variate time series y t ; èt =1; 2;:::;è as conditionally independent, Gaussian random vectors with variance matrices æ t ; denoted by Nèy t j0; æ t è for each t: The variance matrix is stochastically time-varying. Bayesian discounting methods arise from a matrix-variate random walk for the æ t process, resulting in simple sequential updating of inverse Wishart posteriors for inference on æ t as time evolves. Using the notation of West and Harrison è1997, chapter 16è, the time t posterior is of the form pèæ t jd t è = W n,1 t èæ t js t è where D t = fd 0 ; y 1 ;:::;y t g = fd t,1 ; y t g is the sequentially updated information set at time t: Here n t is the degrees of freedom and S t a posterior estimate of æ t ; the posterior harmonic mean. The notation W r,1 èæjsè indicates the inverse Wishart distribution with r degrees of freedom and scale matrix S èsee West and Harrison, as referencedè. The sequence of estimates S t is trivially updated sequentially in time by the forward exponential moving average formula S t =è1, a t ès t,1 + a t y t y 0 t è1è with weight a t = 1=è1 + æn t,1 è based on a discount factor æ: This discount factor lies in è0; 1è; is typically between 0.9 and 1 and will be very close to unity for data at high sampling rates. Having analysed a æxed stretch of data t = 1;:::;n; the sequence of estimates S t is revised by the related backward smoothing formula to incorporate the data at times t +1;:::;n in inference on æ t : Denoting the revised estimate of æ t by S t;n ; the formula is given in terms of inverse variance matrices by the backward recursion S,1 t;n =è1, æès,1 t + æs,1 t+1;n è2è for each t = n, 1;n, 2;:::;1; and starting with S n;n = S n : See West and Harrison è1997, pp è for further details, and the various references by Quintana and coauthors listed above for development and application in econometric ænance. In connection with our analyses of exchange rate time series below, extensions to models in which the time series has a non-zero mean, possibly modelled via a dynamic regression model, is straightforward. Then the forward updating formula è1è is modiæed by replacing the observation y t by the appropriate standardised forecast error, following West and 2

4 Harrison èas referenced aboveè. The details are standard and a side issue here, though the modiæcation is important in developing portfolio allocations below. Though there are ranges of possible models for a non-zero, stochastic mean function that may beofinterest, our analysis is restricted to an assumedly constant mean ç here. In modelling exchange rate returns, this will be a vector of very small elements and the impact of admitting a non-zero mean is small in resulting inferences on the æ t sequence. The impact on portfolio allocations that are mean-dependent are not necessarily small, however, so we maintain the more general model. That is, we assume the model y t ç Nèç; æ t è è3è and that the resulting æltering and smoothing equations from Bayesian discounting analyses are modiæed following West and Harrison è1997, pp è. Future reports will discuss models with mean functions involving stochastic regressions, such as underpin the applied models of Quintana and Putnam è1996è, Quintana, Chopra and Putnam è1995è. 2.2 Component and Factor Structure From very early examples and applications of variance matrix discounting, the principal component or factor structure of the variance matrices æ t has been the subject of analysis in aiding understanding of the nature of changes in covariance patterns, and the underlying latent mechanisms driving such changes. See, for example, the studies of monthly exchange rate time series in Quintana and West è1987è, also reported in West and Harrison è1997, section è, where the patterns of change over time in the principal component structure of the estimates S t;n are explored. A standard principal component decomposition of S t;n provides insight into the related decomposition of æ t : Often, as in the above examples, this will yield a small number of dominant components representing latent factors contributing measurably to both total variability in the series and the covariance structure, together with additional residual components. This underlies the interest in dynamic factor models to more explicitly represent the latent structure and put the spot-light on inference on factor processes and their parameters directly. In line with Jacquier, Polson and Rossi è1994è, Kim, Shephard and Chib è1998è, and èextrapolating to the stochastic volatility context hereè Geweke and Zhou è1996è, a basic k,factor dynamic model èwith kéq;è for æ t is where æ t = X t H t X 0 t + æ t = kx x tj x 0 tj h tj + æ t æ X t is the q æ k factor loadings matrix at time t; with columns x tj ; æ H t = diagèh t1 ;:::;h tk è is the diagonal matrix of instantaneous factor variances, and æ æ t j=1 = diagèè t1 ;:::;è tk è is the diagonal matrix of instantaneous, series-speciæc or ëidiosyncratic" variances. In terms of the time series y t in è3è, this is equivalent to the representation è4è y t = ç + X t f t + æt è5è 3

5 where æ f t ç Nèf t j0; H t è are conditionally independent realisations of the k,vector latent factor process, æ æt ç Nèætj0; æ t è are conditionally independent and series-speciæc quantities, and æ æt and f s are mutually independent for all t; s: Principal component decompositions of estimated variance matrix sequences provide insight into the relevant numbers of ëimportant" factors k; the factor loadings and variance components, and their changes through time. Note that the general factor model above does not, however, necessarily involve the notation of orthogonal factor structure key to principal component methods. 2.3 Factor Model Constraints From here on we restrict attention to models with constant factor loadings and idiosyncratic variances, so that X t = X and æ t = æ for all t: This provides a framework very similar to those mooted by the earlier authors, as referenced above, in which we aim to investigate the issues and diæculties in model implementation. We have, however, implemented models that do allow for changes in time, particularly in the factor loadings, and this will be necessary in some application; more on that in a future article. Well-known questions of model identiæcation and parametrisation arise immediately. First, concerning the number of parameters in the factor model, and second concerning the invariance under invertible linear transformations of the factor vectors. There are many approaches to constraining the model for identiæcation, each raising its own questions of interpretation of the resulting factor structure èpress 1985, chapter 10; Press and Shigemasu 1989è. Our preference here is to adopt a variant of the ëhierarchical" constraints used, for example, in Geweke and Zhou è1996è. Speciæcally, the loadings matrix is given in the form 0 1 X = æææ 0 x 2;1 1 0 æææ 0... æææ 0 x k;1 x k;2 x k;3 æææ 1 x k+1;1 x k+1;2 x k+1;3 æææ x k+1;k... æææ. x q;1 x q;2 x q;3 æææ x q;k C A : è6è One immediate implication is that the chosen order of the univariate time series in the y t vector is viewed as deæning the factors: the ærst series is the ærst factor plus a ënoise" term, and so forth. This focuses attention on the choice of ordering in model speciæcation, and provides interpretation. A second implication relates to the constraint on the number of factors allowed. If all the idiosyncratic variances è j are non-zero, then, simply by counting the number of free parameters in the implied variance matrix æ t ; we deduce an upper bound on k implied by solutions to the quadratic inequality qèq +1è, 2èqk + qè, kèk +1è ç 0. 4

6 For example, with q =6or7wehave k ç 3; with q =15or16wehave k ç 10; while with q =30wehave k ç 22: For realistic values of q this bound is unlikely to be problematic, as practical interest will be in models with smaller numbers of factors. 2.4 Stochastic Volatility for Dynamic Factors Multivariate generalisations of univariate SV models through dynamic factor models are mentioned by various authors, including Harvey, Ruiz and Shephard è1994è, Shephard è1996è, Kim, Shephard and Chib è1998è, and have been investigated by Jacquier, Polson and Rossi è1995è. The basic model of the latter authors assumes that the univariate factor series f ti follow standard univariate SV models, but discuss possible extensions also mentioned in Kim, Shephard and Chib è1998è. We adopt such an extension here, one in which the log volatilities of the factors follows a vector autoregression with possibly correlated innovations. Before proceeding, we note that related developments of multivariate models appear in the multivariate ARCH and related areas, in studies of both SV issues and common components models. Key contributions include Engle è1982è, Bollerslev, Engle and Wooldridge è1988è, Harvey and Stock è1988è, Diebold and Nerlove è1989è, Engle, Ng, and Rothschild è1990è, Bollerslev and Engle è1993è, Míuller and Pole è1994è, King, Sentana and Wadhwani è1994è. Much of what is developed below, in terms of model analysis and inference, would easily transfer to alternative and related models in the areas represented in these various references. For each i = 1;:::;k deæne ç ti = logèh ti è; and write çt = èç t1 ;:::;ç tk è: We assume a stationary vector autoregression of order one, VARè1è, centered around a mean ç = èç 1 ;:::;ç k è 0 and with individual AR parameters ç i in the matrix æ = diagèç 1 ;:::;ç k è: That is, for t =1; 2 :::; we have çt = ç +æèçt,1, çè+!t è7è with independent innovations!t ç Nè!tj0; Uè è8è for some innovations variance matrix U: The implied marginal distribution for each çt is then çt ç Nèçtjç; Wè è9è where W satisæes W = æwæ + U: Note that this marginal variance matrix has elements W ij = U ij =èè1,ç i èè1,ç j èè: The model allows dependencies across volatility series through non-zero oæ-diagonal entries in U and W: Also, this marginal distribution deænes the initial distribution for ç 1 : 3 Bayesian Inference and Computation The model as speciæed so far comprises the basic factor structure è5è with supporting assumptions of conditional normality andindependence, combined with the SV model è7è and its supporting assumptions. Model completion for Bayesian analysis requires prior distributions for the full set of parameters fç; X; æ; ç; æ; Wg: Bayesian inference for any 5

7 speciæed prior requires the computation and summarisation of the implied posteriors for these model parameters, together with inferences on the factor processes ff t ;t=1; 2;:::;ng and the log-volatility sequence fçt; t = 1; 2;:::; ng over the time windowofn consecutive observations comprising the information set D n : Computation via MCMC simulation builds on both the work of previous authors in the SV and factor modelling literature, and previous work in quite diæerent models with related technical structure èaguilar and West 1998; West and Aguilar 1997è. To complete the model speciæcation, we assume a prior speciæed in terms of conditionally independent components pèçèpèxèpèæèpèçèpèæèpèuè è10è where the chosen marginal priors are either standard reference priors or proper priors that are chosen to be conditionally conjugate, as discussed below. The outlook here is to explore the use of reference priors to the extent possible to provide an initial analysis framework. Our prior speciæcations reæect this view, though these models do require the use of informative, proper priors for some model components due to identiæcation issues, as we will discuss. Further, speciæc applications may use alternative prior speciæcations, both in terms of informative priors on model components and in terms of prior dependencies between parameters, though we do not discuss other prior speciæcations here. First, we assume standard reference priors for the univariate entries in the conditional mean, the factor loading matrix and the idiosyncratic variance matrix, so that pèçèpèxèpèæè è qy j=1 è,1 j : Note that the prior for X is, of course, subject to the speciæed 0è1 constraints on values in the the upper triangle and diagonal in è6è, so the constant prior density applies only to the remaining, uncertain elements. Second, we use independent normal priors for the univariate elements of ç and the diagonal elements of æ: This allows for both reference priors, by setting the prior precisions to zero, and restriction of the values of each ç j by adapting the prior to be truncated to è0; 1è: Finally, we use an informative inverse Wishart prior for the VARè1è innovations variance matrix U: This will often be speciæed with hyperparameters based on prior data analysis, as we illustrate below. Notice that an improper reference prior on U; together with that so speciæed for æ; is simply inappropriate, as the two determine separate sources of variability in the data that are confounded in the model. This point, rather critical to model implementation and resulting data analysis, is almost implicit in the prior work of Kim, Shephard and Chib è1998è. These authors use informative proper priors for innovations variances that parallel our assumptions in their univariate SV models; though they present these priors without further discussion, the propriety is critical in overcoming otherwise potentially problematic confounding issues. Hence initial analysis of previous data, or some other prior elicitation activity, is needed. In our applied development below we explore the use of Bayesian variance discounting analyses in providing easy preliminary analysis of a reserved initial section of data as input to this. For now, the key point is that the prior for U is both proper and has the conditionally conjugate inverse Wishart form. Iterative posterior simulation uses an MCMC strategy that extends those in existing SV models èjacquier, Polson and Rossi 1995; Kim, Shephard and Chib 1998è to the multivariate 6

8 case, introduces elements of MCMC algorithms for Bayesian factor analysis as in Geweke and Zhou è1996è, and adds novel components derived from models with latent VAR components developed in a quite diæerent context èaguilar and West 1998; West and Aguilar 1997è. We iteratively simulate values of all model parameters together with the full set of values of the latent processes f t and çt by sequencing through the a set of conditional distributions detailed in the Appendix. At some stages we have direct conditional simulations, at others we require the introduction of novel Metropolis-Hastings acceptèreject steps. We note in passing that, from an algorithmic viewpoint, there are various possible extensions and alternative methods for components of the MCMC analysis, such as in utilising some of the ideas from Shephard and Pitt è1997è for example, though we have not explored such variants yet. Beyond the appendix material here, further technical details are available on request from the authors, as is Fortran software for this implementation. 4 Studies of International Exchange Rates 4.1 Data and Initial Discounting Analyses Figure 1 displays time series graphs of the weekday closing spot exchange rates of several currencies relative to the US dollar during the period 10è09è86 to 08è09è96, a total of 2567 data points in each series. The currencies are, in order, the DeutschmarkèMark èdemè, Japanese Yen èjpyè, Canadian Dollar ècadè, French Franc èfrfè, British Pound ègbpè and Spanish Peseta èespè. We analyse one-day-ahead returns y ti = s ti =s t,1;i, 1 for currency i = 1;:::; q = 6; as graphed in Figure 2. Initial analyses using variance matrix discounting are summarised in Figures 3 and 4. Three separate analyses were run, diæering only through the value of the discount factor, speciæed as æ = 0:9; 0:95; 0:99 for the three cases. The initial prior distribution in each caseisvery vague, namely W,1 1 èæjiè: In each of the three analyses, principal components decompositions were made of each of the posterior estimates S t;n over t = 1;:::;n = 2567: In each analysis and essentially uniformly over the time period, this yields three dominant components with fairly stable time trajectories for the corresponding eigenvectors representing the dynamic factor loadings. Figure 3 displays the time trajectories of the diagonal elements of S t;n ; i.e., the sequence of posterior point estimates of the conditional variances of the six currencies. Figure 4 displays the corresponding related trajectories of the estimates of the variances of the underlying latent factors, namely the eigenvalues of the S t;n matrices over time. The greater adaptivity induced by lower discount factors is apparent in these graphs. The very low æ = 0:9 is over-adaptive, responding very markedly to small changes in realised volatilities. In contrast, the higher discount factor æ = 0:99 induces a much greater degree of smoothing of the volatility trajectories, and is likely under-adaptive in time of really marked change, such as towards the end of 1992 when Britain withdrew from the EU exchange rate agreement, resulting in marked swings and increased volatility in the European currencies across the board. The impact of this event is evident in the estimated trajectories of both the marginal variances of currencies and in the corresponding variances of the factors arising from the direct principal components decompositions in Figure 4. Notice that the end-1992 volatility changes impact across all factors, highlighting the apparent dependencies in factor trajectories across the entire time period. This indicates 7

9 the need for dependence structure in modelling latent volatility processes in dynamic factor analyses, as is allowed in the theoretical framework described above and investigated in factor model data analyses below. We view such dynamic principal component analyses as providing informal, exploratory views of possible latent factor structure, albeit conditioned on the mathematically convenient but practically questionable orthogonality constraints. It appears that at most three factors are necessary, which is anticipated as the currencies represent three distinct trading blocs: Canada, Japan and the EU. The trajectories of the three minor eigenvalues remain at consistently negligible levels across the time frame here, so that a model with three factors plus currency-speciæc random eæects is indicated. We now adopt such a model. 4.2 Dynamic Factor Analysis Taking q = 6 and k = 3 in the dynamic factor model è5è provides a maximal speciæcation: under the assumed structure of the factor loadings matrix è6è, and assuming each of the è j to be non-zero, the number of factors must necessarily be no greater than three. Hence, in addition to being suggested by the discounting analyses, this serves here as an encompassing model; if fewer than three factors are supported by the data, that fact will be reæected in posterior inferences about factor loadings and variances. We base an appropriate, informative prior for the key matrix U in the volatility model on a pre-analysis of the initial 200 observations on the time series, reserving these few observations for this alone and then analysing the remaining data with the factor model. From the Bayesian discount model with a discount factor of 0.9, we extracted the point estimates of the three dominant eigenvalues of each S t;n and used these as ad-hoc estimates of the factor volatilities h tj ; for each j; t: Three separate ARè1è models where then ætted to the log-volatilities so computed, using standard reference Bayesian analyses. This provided posteriors for the AR parameters and innovations variances, in each volatility series marginally, that we take as ëball-park" initial estimates to be used to specify an informative prior for U prior to analysis of the remaining data. This preliminary analysis gave approximate prior means of the three innovations variances around 0.001í With this in mind, we chose the prior for U in the factor model analysis to be inverse Wishart W r,1 0 èujr 0 è with r 0 =100 degrees of freedom èhalf the prior sample size in the ad-hoc analysisè and R 0 =0:0015I; appropriately ëcentering" the prior for U: Note that the prior does not anticipate correlations across volatility processes, though this could easily be done. The MCMC analysis of this factor model involved a range of experiments with Monte Carlo sample sizes and starting values, and MCMC diagnostics. Our summary numerical and graphical inferences are based on over 20,000 simulations of posteriors, generated following a 5,000 burn-in period. We subsample a set of 1,000 spaced 20 apart so as to break correlations and record resulting samples for graphical display purposes. Summary graphs appear in Figures 5 to 9 inclusive. First, Figure 5 graphs estimated trajectories of conditional variances of the currencies í the posterior means of the diagonal elements of æ t = XH t X + æ: Note the similarity with the trajectories from the more adaptive of the discount analyses in Figure 3, as is to be expected. Next, Figure 7 provides histogram approximations to marginal posteriors for the elements of X: Note that the ærst column gives positive weight to all but CAD, representing the relative strength of the US dollar to 8

10 the currencies of the EU and Japan. The CAD has almost no weight here, as to be expected as its value in international markets is most strongly determined by the US dollar alone, and the relative values of the weights on the EU countries naturally reæect their relative strengths. The loadings on the second factor are very small and, if non-negligible, negative, apart from Japan with the æxed unit weight. This is therefore largely the Japan:US factor, with some residual contrast between Japan and the rest of the currencies. Similar comments applies to the third factor which essentially represents the Canadian:US rates, and in which the main residual contrast is that reæecting the diæerential status of Britain to the rest of the EU, presumably driven in part by the departure of Britain from the exchange rate control system. The graphs in Figure 6 display the trajectories of approximate posterior means for the three factor processes f tj and their conditional standard deviations p h tj : The main points to note here are the appearance of peaks in the volatility processes consistent with positive correlations in volatility across the three factors, and the relative scales of volatility: all three factors are evidently contributing measurably to overall variability in the multiple series, though the factors appear to be roughly ordered in terms of decreasing overall levels. Again, this is consistent with expectations from a substantive viewpoint. Figures 8 and 9 summarise marginal posterior inferences for key æxed model parameters, all in boxplot form. Figures 8 displays boxplots of posterior margins as follows. the upper left frame displays margins for the elements of the conditional mean ç: The upper right frame displays margins for the diagonal elements p è j of the residual variance matrix æ: The lower frame displays margins for 100è j =ç 2 tj where ç2 tj is the jth diagonal element ofæ t : These ratios measure percent total variation in each of the currency series that is contributed by the idiosyncratic terms í generally non-negligible, and appreciable for both GBP and ESP. Similar displays appear in Figure 9 for elements of the parameters ç; æ and U in the VARè1è volatility model. Speciæcally, the upper left frame displays margins for the three parameters expèç j =2è where the ç j are the entries of the stationary mean ç of the VARè1è model. Converting from the log-volatility tovolatility scales, these scale factors expèç j =2è represent the standard deviations of the implied stationary distribution, i.e., base levels of conditional variation in the three factor processes. The rough ordering of factors according to marginal variability is clear here. The upper right frame displays margins for the AR parameters ç j in ç; indicating that all three are obviously very close to, but less than, unity, and so representing high persistence in the volatility processes. The approximate posterior means for the three ç j are 0.97, 0.98 and 0.98, respectively. The lower frame displays margins for the standard deviations and correlations of matrix W; the marginal variance matrix in the VARè1è volatility model. The earlier noted positive correlations between factor processes are indicated here. Related numerical summaries provide approximate posterior means of the variances in W as 0.50, 0.86 and 0.82, respectively, while the corresponding posterior means for the variances in the innovations matrix U are 0.027, and 0.025, respectively. 9

11 4.3 Comparison of Models with Constrained Portfolio Allocations Model comparisons are made with explicit focus on one-step forecast accuracy in the context of dynamic portfolio allocations, essentially following the perspective of Quintana è1992è, Putnam and Quintana è1994è, and Quintana and Putnam è1996è. A similar perspective is adopted in Polson and Tew è1997è though with very diæerent models. Our comparisons are based on posterior distributions from the models ætted to the entire time series, so that they do not represent real-time, sequential forecasts, but nevertheless do provide a coherent basis for model comparisons with a utility function directly measuring real-world performance in terms of cumulative ænancial return. In this section, we adopt traditional portfolios in which the total sum invested at each time point is æxed, focusing mainly on questions about how the discount and factor models diæer in terms of resulting cumulative returns. At each time point t, 1; we suppose that an existing investment in the various currencies under study may be reallocated according to a portfolio a t for the next time point. The elements of a t are the $US amounts invested in the corresponding currency. For this comparative analysis, we assume no transaction costs and that we may freely reallocate dollars instantaneously to long or short positions across the currencies, subject only a 0 t1 =1: The realised portfolio return at time t is the $US amount r t = a 0 ty t ; and models may be compared on the basis of cumulative returns over chosen time intervals. The portfolio allocation decision problem involves the general Markowitz mean-variance optimisation, and we apply this at each time point one-step ahead. In all models, the time t situation is summarised through posterior one-step ahead means and variance matrices for y t ; denoted here by g t and G t : The decision context targets minimisation of the one-step ahead variance of returns subject to speciæed target means. For a speciæed target mean return m; the one-step portfolio a t will be chosen to minimise the one-step ahead variance of returns a 0 tg t a t subject to constraints a 0 t1 =1anda 0 tg t = m: The well-known solution is where with a èmè t = G,1 t èag t + b1è a = 1 0 G,1 t e and b =,g 0 tg,1 t e e =è1m, g t è=d and d =è1 0 G,1 t 1èèg 0 tg,1 t g t è, è1 0 G,1 t g t è 2 : Compared to this are two other standard portfolio allocations: the target-independent allocation derived at the boundary of the mean-variance eæcient frontier, namely a èmeè t =è1 0 G,1 t g t è,1 G,1 t g t ; and the strictly risk-averse minimum variance portfolio a èmvè t =è1 0 G,1 t 1è,1 G,1 t 1: Figure 10 graphs the trajectories of cumulative returns over time based on these four models in the frames in the ærst two rows. These four frames correspond to four diæerent æxed investment strategies: the two risk-averse strategies a èmvè t and a èmeè t and then two 10

12 strategies a èmè t with daily target returns m = 0:00016 and m = 0:00028 respectively. These appear in the order ètop leftè a èmvè t ; ètop rightè a èmeè t ; ècenter leftè a è0:00016è t and ècenter rightè a è0:00028è t : After a period in which the model behave very similarly in the determination of portfolio allocations, the changes in volatility in 1992 are more appropriately captured by the most adaptive discount model èæ = 0:9è and the dynamic factor model. These two models proceed to generally dominate the others in terms of cumulative returns under the practically relevant strategies. The factor model clearly wins across all cases in this time window. The trajectories of optimal portfolio weights for the target-independent case appear in Figure 11. This displays the changing values of a èmeè t from the three Bayesian discount analyses and also from the dynamic factor model analysis. The weights are graphed as percentages, i.e., simply 100 times their actual values, as they are constrained to sum to unity. It is interesting to note that, though the very adaptive discount model produces cumulative returns that closely shadow the factor model, the weights in the factor model are relatively much more stable over time. The discount model adapts the weights quite widely as it permits very marked patterns of change in the full variance matrix of returns, whereas the factor model assigns changes in observed volatilities to appropriate model components and so induces more stability inweight trajectories. 4.4 Investment Performance with Unconstrained Portfolios We now move to a further portfolio allocation study that is focused on comparison of portfolio strategies rather than on comparison of models. Here we consider allocations in which the portfolios are completely unconstrained; that is, we remove the unit sum constraint on the allocation vector a t : This means that we maychoose each allocation without regard to resources, permitting arbitrary long or short positions across the currencies. This typiæes the practical working context in global investments in large ænancial institutions, and is in line with recent work with discount models èquintana and Putnam 1996è. Under an unconstrained strategy, the optimum allocation at time t is given by where a èæmè t = ag,1 t g t a = m=g 0 tg,1 t g t : The third row of graphs in Figure 10 displays the cumulative returns trajectories using the optimal portfolio weights a èæmè t from this analysis, to be compared to the earlier ægures using the constrained portfolios. The two graphs provide displays under portfolios èlower leftè a èæ0:00016è t and èlower rightè a èæ0:00028è t : Now we see that the dynamic factor model is very clearly dominant, achieving cumulative returns that are about twice as large as those of the most competitive discount model, and, by the way, also exceeding those of the constrained models. The response following the major structural changes in volatility at the end of 1992 leads to a marked swing in portfolio structure that the unconstrained allocations capitalise on in a major way in the factor model, with a persistent eæect on cumulative returns thereafter. The trajectories of optimal, unconstrained portfolio weights for the case m = 0:00016 are graphed in Figure 13. The values plotted are 100 times the actual weights divided 11

13 at each time point, indicating the relative weight of each currency in the portfolio at each time. This provides a direct comparison with the corresponding trajectories of weights from the unit-sum constrained allocation appearing in Figure 12, where the actual and relative values coincide. The impact of the Britain's withdrawal from the EU exchange rate agreement, in late 1992, and the resulting marked increases in volatility and the portfolio's response, are very clear in Figures 13 and 10. The allocations shift swiftly and quite radically in extent to short positions on Sterling and the strongly associate Peseta, while simultaneously adopting radically long positions on the strong Mark and Yen. At the same time, the total investment dropped markedly; the major changes by the total 1 0 a èæmè t in volatility led to the anticipation of high levels of increased risk, and the total 1 0 a èæmè t invested in the market decreased radically as a result. Figure 14 graphs the time trajectory of the total 1 0 a èæmè t ; indicating relative stability in the æuctuations around a level of unity, but with marked swings up and down in periods of low and high volatility, respectively. 5 Concluding Comments Our investigations indicate the feasibility of formal Bayesian analysis of structured dynamic factor models. The analysis is accessible computationally with nowadays moderate computational resources, and our empirical studies suggest that the analysis will be manageable with dimensional time series and several factors. We are currently investigating more extensive applications in short-term forecasting and on-line portfolio allocations with higher dimensional models for longer-term exchange rate futures. The example here is suggestive of potential beneæts, and supportive of the view that exploiting systematic volatility patterns via factor structuring may yield very substantial improvements in short-term forecasting and decision making in dynamic portfolio allocation, especially in the unconstrained optimisation as illustrated in the ænal row of graphs in Figure 10. In the case of constrained portfolio optimisations, the over-adaptive discounting method does reasonably well at times, though is clearly eventually dominated in terms of cumulative return trajectories by the factor model. We conjecture that, in studies of forecasting and portfolio allocation with longer term horizons, such as 30-day exchange rate futures, and in extended models that incorporate dynamic regression components, the dominance of the factor modelling approach will be even clearer. This is the subject of current and near-future research. The dynamic factor models illustrated are amenable to direct implementation using our customised MCMC methods with the minimalèreference prior speciæcations we have used here. Our use of the variance discounting method on a reserved initial section of the data to provide input to informative priors is important in identifying ëball-park" scales for the U matrix of the VARè1è SV model. Though not pursued here, other aspects of such preliminary analyses may be used to determine informative priors for other elements of the factor model. The established discounting methods are, relative to dynamic factor models, trivial to implement in the current context, a fact that is important in using discount methods to specify partial prior structure in the dynamic models. Our empirical ændings indicate that, not surprisingly with this kind of data, moderately adaptive discount methods fare well in time of slow change in volatility levels and patterns, but are relatively uncompetitive in cases of more marked structural change. This is to be expected. Looking ahead, 12

14 models and approaches that attempt to simplify the process of factor modelling, perhaps somehow integrating elements and concepts of variance matrix discounting into a speciæed factor structure, may be attractive from a computationalèimplementation viewpoint. We are currently investigating such a synthesis of approaches. In the factor model context per se, there are several relevant technical and modelling issues to be explored. These include questions of choices about the numbers of factors, and about the ordering of time series in the context of the speciæc structure adopted for factor loadings. Model extensions under investigation relax the assumptions of constancy of the factor loadings, and we anticipate extensions to models in which the conditional mean is a dynamic regression, as is likely very necessary for serious practical application. In the meantime, further study and empirical assessments on time series with larger numbers of univariate components and larger numbers of factors are under investigation. Our experience to date leads us to believe that such investigations will be every fruitful and support the preliminary conclusions reached in this report about the potential utility of factor models. Appendix In each of the following subsections we detail the conditional posteriors for various parameters and latent variables in turn. At each step it is implicit that we are conditioning on æxed values èpreviously simulated valuesè of all other variables. As noted in the text, further technical details are available on request from the authors, as is Fortran software for this implementation. Sampling the conditional mean The basic model è3è and the uniform prior for ç immediately imply a multivariate normal conditional posterior for ç given the values of æ t : This is easily sampled. Sampling latent factors The full conditional distribution of f t is given by Nèf t ja t èy t, çè; H t, A t Q t A 0 tè where Q t = XH t X 0 + æ and A t = H t X 0 Q,1 t : The f t are conditionally independent and so sample values are drawn independently from this set of normal distributions for t = 1; 2 :::;n: Sampling factor loadings From the model, the conditional likelihood function for the factor loading matrix X is Q nt=1 Nèy t, çjxf t ; æè: This is a standard form, log-quadratic in the uncertain elements of X; and so combines with a normal or uniform reference prior to imply a multivariate normal conditional posterior, which is easily sampled. 13

15 Sampling idiosyncratic variances The elements of æ are conditionally independent with inverse gamma distributions. Let g i = P n t=1 èy ti, ç i, z 0 if t è 2 where z i is the i th row of X: Then è i has the inverse gamma distribution with shape n=2 and scale g i =2; denoted by Gèè,1 i Sampling the mean log-volatilities jn=2;g i =2è: Under a normal prior Nèçjm 0 ; M 0 è; the conditional posterior is normal Nèçjm; Mè with and M,1 = M,1 0 + W,1 +èn, 1èèI, æèu,1 èi, æè Mm = M,1 0 m 0 + W,1 ç 1 +èi, æèu,1 nx t=2 èçt, æçt,1è: The case of a uniform reference prior is recovered by setting M,1 0 = 0: Sampling the VAR coeæcients The structure of the conditional posterior for æ; and the resulting Metropolis-Hastings strategy for simulation, is precisely as developed for component VAR models in a quite diæerent context in West and Aguilar è1998è and Aguilar and West è1997è. Writing æ t = çt, ç; note that the full conditional posterior density for æ is proportional to pèæènèæ 1 j0; Wè ny t=2 Nèæ t jææ t,1 ; Uè where W = æwæ+u is easily evaluated as a function of æ and U: Write ç =èç 1 ;:::;ç k è 0 for the diagonal of æ; and E = diagèæ t,1 è: Then the conditional posterior may be written as proportional to pèæècèæèn èçjb; Bè where and B,1 = nx E 0 U,1 E and Bb = t=2 nx E 0 U,1 æ t t=2 cèæè =jwj,i=2 expè,traceèw,1 æ 1 æ 0 1 è=2è: Under independent normal or uniform priors for the ç j ; the full conditional posterior distribution for æ is the multivariate normal Nèçjb; Bè truncated to the è0; 1è regions in each dimension, and then multiplied by the factor cèæè: This may be sampled by several methods. We use a Metropolis Hastings algorithm that takes the truncated multivariate normal component as proposal distribution. That is, given a ëcurrent" value of ç; with corresponding matrices æ and W; we sample a ëcandidate" vector ç æ from this truncated normal, compute the corresponding diagonal matrix æ æ and variance matrix W æ such that W æ = æ æ W æ æ æ + U; then accept this new ç vector with probability minf1;cèæ æ è=cèæèg: 14

16 Sampling the VAR innovations variance matrix Again, the structure of the conditional posterior for the innovations variance matrix U of the VARè1è volatility model is closely related to developments in a quite diæerent context in West and Aguilar è1998è and Aguilar and West è1997è. Again using centred volatilities æ t = çt, ç for each t; we have a full conditional posterior for U proportional to where and pèuèaèuèjuj,èn,1è=2 expè,traceèu,1 Gèè G = nx t=2 èæ t, ææ t,1 èèæ t, ææ t,1 è 0 aèuè =jwj,1=2 expè,traceèw,1 æ 1 æ 0 1 è=2è with W = æwæ + U: Under a speciæed inverse Wishart prior W r,1 0 èujr 0 è; this posterior density is proportional to aèuèw r,1 èujrè where r = r 0 + n, 1 and rr = r 0 R 0 +èn, 1èG: We use this inverse Wishart distribution as a proposal distribution in the Metropolis-Hastings algorithm. That is, given a ëcurrent" value of U and corresponding W; we sample a ëcandidate" value U æ from W r,1 èujrè; and accept it with probability minf1;aèu æ è=aèuèg where W æ = æw æ æ + U æ. Sampling the actual latent log-volatilities Given currently imputed values for the full factor process f t over time t; we follow Kim, Shephard and Chib è1998è in transforming each of the univariate time series of factor elements into a non-gaussian linear model form. Speciæcally, foreach factor i and time t deæne z ti =logèfti 2 è and note that z ti = ç ti + ç ti where the ç ti terms are independent and distributed as log,ç 2 1 : In vector form for all k factor series, we thenhave z t = çt + çt where çt = èç t1 ;:::;ç tk è 0 : Based on the current values of the z t ; for each t; this provides the set observation equations for the dynamic linear model with state equations çt = ç + æèçt,1, çè+!t for each t: This is a direct multivariate extension of the univariate approach in Kim, Shephard and Chib è1998è, whose ensuing analysis uses the now established approximation to the distribution of the error terms ç it as a speciæed ænite mixture of normals èshephard 1994bè. The multivariate extension is immediate: 15

17 æ Introduce a set of indicator variables s ti such that that s ti identiæes the normal mixture component for ç ti : æ Conditional on these indicators, we have a multivariate dynamic linear model for the sequence of log-volatility vectors. The forward-æltering, backward-sampling algorithm for state space models ècarter and Kohn 1994; Fríuhwirth-Schnatter 1994; West and Harrison 1997, chapter 15è now applies to directly simulate the full set of vectors fçt;t= 1;:::;ng from the implied conditional posterior. ènote that, in more elaborate models for the volatility processes, the alternative sampling method using the simulation smoother of de Jong and Shephard è1995è may have computational advantages not realised in this, the simplest VAR model.è æ Given these sampled values of the çt; the complete conditional multinomial posterior probabilities over values of the indicators s it are easily computed, the indicators being conditionally independent and so easily sampled. References Aguilar O. and West M. è1998è, ëanalysis of Hospital Quality Monitors using Hierarchical Time Series Models," In Case Studies Bayesian Statistics in Science and Technology: Case Studies 4, èeds: C. Gatsonis et alè, Springer-Verlag, New York èforthcomingè. Ameen, J.R., and Harrison, P.J. è1995è, ënormal Discount Bayesian Models," In Bayesian Statistics 2, èeds: J.M. Bernardo, M.H. De Groot, D.V. Lindley and A.F.M. Smithè, North Holland, Amsterdam, and Valencia University Press. Bollerslev T., Engle R.F. and Wooldridge J.M. è1988è, ëa Capital Asset Pricing Model With Time Varying Covariances," Journal of Political Economy, 96, Carter, C., and Kohn, R. è1994è, ëon Gibbs Sampling for State Space Models," Biometrika, 81, 541í553. Fríuhwirth-Schnatter, S. è1994è, ëdata Augmentation and Dynamic Linear Models," Journal of Time Series Analysis, 15, 183í202. de Jong, P., and Shephard, N. è1995è, ëthe Simulation Smoother for Time Series Models," Biometrika, 82, Diebold F.X. and Nerlove M. è1989è, ëthe Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor ARCH Model," Journal of Applied Econometrics, 4, Engle R.F. è1982è, ëautoregressive Conditional Heteroskedasticity with Estimates of the Variance of the United Kingdom Inæation," Econometrica, 50, Engle R.F., Ng V. K. and Rothschild M. è1990è, ëasset Pricing with a Factor-ARCH Covariance Structure, Empirical Estimates for Treasury Bills," Journal of Econometrics, 45, Geweke J. and Zhou G. è1996è, ëmeasuring the Pricing Error of the Arbitrage Pricing Theory," The Review of ænancial Studies, Vol 9, No 2, Harrison, P.J., and West, M. è1987è, ëpractical Bayesian Forecasting," The Statistician, 36, Harvey A.C. and Stock J.H. è1988è, ëcontinuous Time Autoregressive Models with Common Stochastic Trends," Journal of Economic Dynamics and Control, 12,

18 Harvey A.C., Ruiz E. and Shephard N. è1994è, ëmultivariate Stochastic Variance Models," Review of Economic Studies, 61, Harvey A. C. and Shephard N. è1996è, ëestimation of an Asymmetric Stochastic Volatility Model for Asset Returns," Journal of Business and Economic Statistics, 14, Jacquier E., Polson N.G. and Rossi P. è1994è, ëbayesian Analysis of Stochastic Volatility Models èwith Discussionè," Journal of Business and Economic Statistics, 12, Jacquier E., Polson N.G. and Rossi P. è1995è, ëmodels and Priors for Multivariate Stochastic Volatility," Discussion Paper, Graduate School of Business, University of Chicago Kim S., Shephard N. and Chib S.è1998è, ëstochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, forthcoming. King M., Sentana E. and Wadhwani S. è1994è, ëvolatility and Links Between National Stock Markets," Econometrica, 62, Míuller P. and Pole A. è1994è, ëmonte Carlo Posterior Integration in GARCH Models," Sankhya, èser Bè, to appear. Ng V.K., Engle R.F. and Rothschild M. è1992è, ëa Multi-Dynamic-Factor Model for Stock Returns," Journal of Econometrics, 52, Polson, N.G. and Tew, B.V. è1997è, ëbayesian Porfolio Selection: An Empirical Analysis of the SP500 Index ," Technical Report, Graduate School of Business, University of CHicago. Press, S.J. è1985è, Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference, California: Krieger. Press S.J. and Shigemasu K. è1989è, ëbayesian Inference in Factor Analysis," In contributions to Probability and Statistics: Essays in Honor of Ingram Olkin. eds L.J Gleser, M.D. Perlman, S.J. Press and A.R. Sampson, New York: Springer Verlag. Putnam, B. H. and Quintana, J. M. è1994è, ënew Bayesian Statistical Approaches to Estimating and Evaluating Models of Exchange Rates Determination," In Proceedings of the ASA Section on Bayesian Statistical Science, 1994 Joint Statistical Meetings, American Statistical Association Putnam, B. H. and Quintana, J. M. è1994è, ëshrink Wrap for Currencies," Asset Allocation: Applying Quantitative Discipline to Asset Allocation, èed: Putnam, B.H.è, London: Global Investor, Euromoney Publications, Putnam, B. H. and Quintana, J. M. è1995è, ëthe Evolution of Bayesian Forecasting Models," Asset Allocation: Applying Quantitative Discipline to Asset Allocation, èed: Putnam, B.H.è, London: Global Investor, Euromoney Publications, Quintana, J.M. and West, M. è1987è, ëan Analysis of International Exchange Rates using Multivariate DLMS," The Statistician, 36, Quintana J.M. and West, M. è1988è, ëtime Series Analysis of Compositional Data," In Bayesian Statistics 3, èeds: J.M. Bernardo, M.H. De Groot, D.V. Lindley and A.F.M. Smithè, Oxford University Press. Quintana, J.M. è1992è, Optimal Portfolios of Forward Currency Contracts. In Bayesian Statistics 4, èeds: J.O. Berger, J.M. Bernardo, A.P. Dawid and A.F.M. Smithè, Oxford University Press. Quintana J.M., Chopra V.K. and Putnam B.H. è1995è, ëglobal Asset Allocation: Stretching Returns by Shrinking Forecasts," In Proceedings of the ASA Section on Bayesian Statistical Science, 1995, Joint Statistical Meetings, American Statistical Association. 17

19 Quintana, J. M., and Putnam, B. H. è1996è, ëdebating Currency Markets Eæciency using Multiple-Factor Models," In Proceedings of the ASA Section on Bayesian Statistical Science, 1996 Joint Statistical Meetings, American Statistical Association Shephard N. è1994aè, ëlocal Scale Models: State Space Alternatives to Integrated GARCH Processes," Journal of Econometrics, 6, Shephard N. è1994bè, ëpartial Non-Gaussian State Space," Biometrika, 81, Shephard N. è1996è, ëstatistical Aspects of ARCH and Stochastic Volatility," In D.R. Cox, O.E. Barndoræ-Nielson and D.V. Hinkley eds. Time Series Models in Econometrics, ænance and Other Fields, pp London: Chapman & Hall. Shephard, N., and Pitt, M.K. è1997è, ëlikelihood analysis of non-gaussian measurement time series," Biometrika, 84, Uhlig, H. è1994è, ëon singular Wishart and singular multivariate beta distributions," Ann. Statist., 22, Uhlig, H. è1997è, ëbayesian vector-autoregressions with stochastic volatility," Econometrica, 65, West M. and Aguilar O. è1997è, ëstudies of Quality Monitor Time Series: The V.A. Hospital System". Report for the VA Management Science Group, Bedford, MA., Institute of Statistics and Decision Sciences, Discussion Paper West M., and Harrison, P.J. è1997è, Bayesian Forecasting and Dynamic Models, è2nd Edn.è, New York: Springer Verlag. 18

20 DEM JPY USD/DEM USD/JPY /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 CAD FRF USD/CAD USD/FRF /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 GBP ESP USD/GBP USD/ESP /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 Figure 1: Exchange rate time series. DEM JPY Returns Returns /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 CAD 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 FRF Returns Returns /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 GBP 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 ESP Returns Returns /86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 10/86 9/87 9/88 8/89 8/90 7/91 7/92 6/93 6/94 5/95 5/96 Figure 2: Exchange rate returns. 19